The classical stochastic control theory appears with the birth of stochastic analysis and developed rapidly in recent decades due to its wide range of applications (see Yong–Zhou [1]}), and one of its important applications is the optimal consumption-investment problem under stochastic differential utility (SDU, for short), which was put forward by Duffie–Epstein [2]} in a conditional expectation form. Actually, the SDU is equivalent to the nonlinear backward stochastic differential equations (BSDEs, for short) and its control problem is the stochastic recursive control problem introduced firstly by Peng [3]} and developed by Buckdahn–Li [4]}, Wu–Yu [5]}, Li–Peng [6]}, Pu–Zhang [7]}, Zhou–Dong–Pu [8]}, to name but a few.
Thanks to the importance of BSDEs in modern mathematical finance (see El Karoui–Peng–Quenez [9]}), both stochastic recursive control theory and the optimal consumption-investment problem of stochastic differential utilities achieved great progresses in the past few decades.
